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Distributed delays in a hybrid model of tumor-Immune system interplay
Approximate smooth solutions of a mathematical model for the activation and clonal expansion of T cells
1. | Department of Mathematics and Informatics, University of Messina, Viale F. Stagno d'Alcontres n.31, 98166 Messina, Italy, Italy |
2. | Department I.C.I.E.A.M.A., University of Messina, Contrada Di Dio (S.Agata), 98166 Messina, Italy |
References:
[1] |
M. Dolfin and D. Criaco, A phenomenological approach to the dynamics of activation and clonal expansion of T cells, Mathematical and Computer Modelling, 53 (2011), 314-329. |
[2] |
G. Boillat, "La Propagation des Ondes," $1^{st}$ edition, Gauthier-Villars, Paris, 1965. |
[3] |
G. Boillat, Ondes asymptotiques nonlineaires, Annali di Matematica Pura ed Applicata, IV, CXI (1976), 31-44 (in french). |
[4] |
D. Fusco, Onde non lineari dispersive e dissipative, Bollettino U.M.I, 16-A (1976), 450-458 (in italian). |
[5] |
A. Jeffrey and T. Taniuti, "Nonlinear Wave Propagation," $1^{st}$ edition, Academic Press, New York, 1964. |
[6] |
A. Jeffrey, The propagation of weak discontinuities in quasilinear symmetric hyperbolic system, Z. A. M. P. 14 (1963), 31-314. |
[7] |
A. Jeffrey and T. Kakutani, Weak nonlinear dispersive waves: A discussion centered around the Korteweg-de Vries equation, SIAM Review, 14 (1972), 582-653. |
[8] |
A. Jeffrey, The development of jump discontinuities in nonlinear hyperbolic systems of equations in two independent variables, Arch. Rational Mech. Anal., 14 (1963), 27-37. |
[9] |
P. D. Lax, Nonlinear hyperbolic equations, Comm. Pure Appl. Math., 6 (1983), 231-238. |
[10] |
A. Georgescu, "Asymptotic Treatment of Differential Equations," $1^{st}$ edition, Chapman and Hall, London, 1995. |
[11] |
A. Donato and A. M. Greco, "Metodi Qualitativi per Onde Non Lineari - Quaderni del C. N. R., Gruppo Nazionale di Fisica Matematica, 11th Scuola Estiva di Fisica Matematica, Ravello, (1986), 8-20 September," $1^{st}$ edition, C. N. R. Press, Rome, 1987. |
[12] |
Y. Choquet-Bruhat, Ondes asymptotiques et approchees pour systemes d'equations aux derivees partielles nonlineaires, J. Math. Pures et Appl., 48 (1968), 117-158 (in french). |
[13] |
P. D. Lax, "Contributions to the Theory of Partial Differential Equations," $1^{st}$ edition, Princeton University Press, Princeton, 1954. |
[14] |
P. D. Lax, Hyperbolic systems of conservation law (II), Comm. Pure Appl. Math., 10 (1957), 537-566.
doi: 10.1002/cpa.3160100406. |
[15] |
T. Taniuti and C. C. Wei, Reductive pertubation method in nonlinear wave propagation, J. Phys. Soc. Japan, 24 (1968), 941-946. |
[16] |
V. Ciancio and L. Restuccia, Nonlinear dissipative waves in viscoanelastic media, Physica A, 132 (1985), 606-616. |
[17] |
V. Ciancio and L. Restuccia, Asymptotic waves in anelastic media without memory (Maxwell media), Physica A, 131 (1985), 251-262.
doi: 10.1016/0378-4371(85)90090-1. |
[18] |
V. Ciancio and L. Restuccia, The generalized Burgers equation in viscoanelastic media with memory, Physica A, 142 (1987), 309-320. |
[19] |
A. Jeffrey, "Quasilinear Hyperbolic Systems and Waves," $1^{st}$ edition, Pitman, London, 1976. |
[20] |
I. Muller, "Thermodynamics," $1^{st}$ edition, Pitman Advanced Publishing Program, Princeton, 1985. |
[21] |
I. Muller and T. Ruggeri, "Rational Extended Thermodynamics," $1^{st}$ edition, Springer, Berlin-Heidelberg-New York, 1998. |
[22] |
R. M. Ford and D. A. Lauffenburger, Analysis of chemotactic bacterial distributions in population migraton assays using a mathematical model applicable to steep ar shallow attractant gradients, Bullettin of Mathematical Biology, 53 (1991), 721-749. |
[23] |
D. A. Lauffenburger and K. H. Keller, A Effects of leukocyte random motility and chemotaxis in tissue inflammatory response, Theoretical Biology, 81 (1979,) 475-503. |
[24] |
A. Tosin, D. Ambrosi and L. Preziosi, Mechanics and chemotaxis in the morphogenesis of vascular networks, Mathematical Biology, 68 (2006), 1819-1836.
doi: 10.1007/s11538-006-9071-2. |
[25] |
H. Byrne and L. Preziosi, Modelling solid tumour growth using the theory of mixtures, Mathematical Medicine and Biology, 20 (2003), 341-366.
doi: 10.1093/imammb/20.4.341. |
[26] |
G. Carini, "Lezioni di Istituzioni di Fisica Matematica," edition, Springer, Mediterranean Press , 1989. |
[27] |
J. D. Murray, "Mathematical Biology, vol I," $2^{nd}$ edition, Springer, Berlin-Heidelberg-New York, 2002. |
[28] |
J. D. Murray, "Mathematical Biology, vol II," $2^{nd}$ edition, Springer, Berlin-Heidelberg-New York, 2002. |
[29] |
E. Hopf, The partial differential equation ut + uux = xx, Comm. Appl. Math., 3 (1950), 201-230. |
show all references
References:
[1] |
M. Dolfin and D. Criaco, A phenomenological approach to the dynamics of activation and clonal expansion of T cells, Mathematical and Computer Modelling, 53 (2011), 314-329. |
[2] |
G. Boillat, "La Propagation des Ondes," $1^{st}$ edition, Gauthier-Villars, Paris, 1965. |
[3] |
G. Boillat, Ondes asymptotiques nonlineaires, Annali di Matematica Pura ed Applicata, IV, CXI (1976), 31-44 (in french). |
[4] |
D. Fusco, Onde non lineari dispersive e dissipative, Bollettino U.M.I, 16-A (1976), 450-458 (in italian). |
[5] |
A. Jeffrey and T. Taniuti, "Nonlinear Wave Propagation," $1^{st}$ edition, Academic Press, New York, 1964. |
[6] |
A. Jeffrey, The propagation of weak discontinuities in quasilinear symmetric hyperbolic system, Z. A. M. P. 14 (1963), 31-314. |
[7] |
A. Jeffrey and T. Kakutani, Weak nonlinear dispersive waves: A discussion centered around the Korteweg-de Vries equation, SIAM Review, 14 (1972), 582-653. |
[8] |
A. Jeffrey, The development of jump discontinuities in nonlinear hyperbolic systems of equations in two independent variables, Arch. Rational Mech. Anal., 14 (1963), 27-37. |
[9] |
P. D. Lax, Nonlinear hyperbolic equations, Comm. Pure Appl. Math., 6 (1983), 231-238. |
[10] |
A. Georgescu, "Asymptotic Treatment of Differential Equations," $1^{st}$ edition, Chapman and Hall, London, 1995. |
[11] |
A. Donato and A. M. Greco, "Metodi Qualitativi per Onde Non Lineari - Quaderni del C. N. R., Gruppo Nazionale di Fisica Matematica, 11th Scuola Estiva di Fisica Matematica, Ravello, (1986), 8-20 September," $1^{st}$ edition, C. N. R. Press, Rome, 1987. |
[12] |
Y. Choquet-Bruhat, Ondes asymptotiques et approchees pour systemes d'equations aux derivees partielles nonlineaires, J. Math. Pures et Appl., 48 (1968), 117-158 (in french). |
[13] |
P. D. Lax, "Contributions to the Theory of Partial Differential Equations," $1^{st}$ edition, Princeton University Press, Princeton, 1954. |
[14] |
P. D. Lax, Hyperbolic systems of conservation law (II), Comm. Pure Appl. Math., 10 (1957), 537-566.
doi: 10.1002/cpa.3160100406. |
[15] |
T. Taniuti and C. C. Wei, Reductive pertubation method in nonlinear wave propagation, J. Phys. Soc. Japan, 24 (1968), 941-946. |
[16] |
V. Ciancio and L. Restuccia, Nonlinear dissipative waves in viscoanelastic media, Physica A, 132 (1985), 606-616. |
[17] |
V. Ciancio and L. Restuccia, Asymptotic waves in anelastic media without memory (Maxwell media), Physica A, 131 (1985), 251-262.
doi: 10.1016/0378-4371(85)90090-1. |
[18] |
V. Ciancio and L. Restuccia, The generalized Burgers equation in viscoanelastic media with memory, Physica A, 142 (1987), 309-320. |
[19] |
A. Jeffrey, "Quasilinear Hyperbolic Systems and Waves," $1^{st}$ edition, Pitman, London, 1976. |
[20] |
I. Muller, "Thermodynamics," $1^{st}$ edition, Pitman Advanced Publishing Program, Princeton, 1985. |
[21] |
I. Muller and T. Ruggeri, "Rational Extended Thermodynamics," $1^{st}$ edition, Springer, Berlin-Heidelberg-New York, 1998. |
[22] |
R. M. Ford and D. A. Lauffenburger, Analysis of chemotactic bacterial distributions in population migraton assays using a mathematical model applicable to steep ar shallow attractant gradients, Bullettin of Mathematical Biology, 53 (1991), 721-749. |
[23] |
D. A. Lauffenburger and K. H. Keller, A Effects of leukocyte random motility and chemotaxis in tissue inflammatory response, Theoretical Biology, 81 (1979,) 475-503. |
[24] |
A. Tosin, D. Ambrosi and L. Preziosi, Mechanics and chemotaxis in the morphogenesis of vascular networks, Mathematical Biology, 68 (2006), 1819-1836.
doi: 10.1007/s11538-006-9071-2. |
[25] |
H. Byrne and L. Preziosi, Modelling solid tumour growth using the theory of mixtures, Mathematical Medicine and Biology, 20 (2003), 341-366.
doi: 10.1093/imammb/20.4.341. |
[26] |
G. Carini, "Lezioni di Istituzioni di Fisica Matematica," edition, Springer, Mediterranean Press , 1989. |
[27] |
J. D. Murray, "Mathematical Biology, vol I," $2^{nd}$ edition, Springer, Berlin-Heidelberg-New York, 2002. |
[28] |
J. D. Murray, "Mathematical Biology, vol II," $2^{nd}$ edition, Springer, Berlin-Heidelberg-New York, 2002. |
[29] |
E. Hopf, The partial differential equation ut + uux = xx, Comm. Appl. Math., 3 (1950), 201-230. |
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